# 2023 Tarski Lecture

Richard Shore was on the faculty at Cornell University from 1974 to

2020 at which point he retired as Goldwin Smith Professor of Mathematics

Emeritus. During those years he held visiting appointments at Harvard, The

Hebrew University, M.I.T., MSRI (Berkeley), National University of Singa-

pore, The University of Chicago, University of Sienna and others institutions.

In addition to his personal research grants, Shore has been a principal in-

vestigator for binational grants with mathematicians in Greece, Israel, Italy

and New Zealand.

Shore has been a speaker at the ICM (1983) and the ICLMPS (1999) as

well as many other venues. He was the Gödel Lecturer for the ASL in 2009.

He has served the ASL in many capacities including President (2001-04).

Since 2008, he has been the Publisher of the ASL. Among his other editorial

work, Shore was the founding managing editor of the Bulletin of Symbolic

Logic (1993-2000). He was also a member of the board of Project Euclid,

an early online mathematical publishing platform, from its launch in 2002

until 2013. Among the eighteen thesis students he supervised, four (with one

as secondary advisor) have won the Sacks Prize for the year's best thesis in

logic.

**Lecture 1 of 2 **(April 24, 2023 @ 4:10–5:00 pm, UC Berkeley Campus, Location: Physics 3)**: **

Mathematics: A Multiverse: The basic question that reverse mathematics attempts to address is how hard is it to prove particular theorems or carry out particular constructions of ordinary classical mathematics. Hard not in the sense of how many hours, days or years or cups of coffee it took a mathematician or many mathemati-cians to prove the theorem but hard in some more mathematical sense. This talk will be an introduction to the subject with hints of a new view. We primarily consider two related standard criteria of hardness and the relations between. One is based on the strength of formal systems of (second order) arithmetic in which theorems can be proved. The second is recursion theoretic and based on the computational complexity of the objects the theorems assert exist. We will also briefly discuss relations to common philosophical approaches to the foundations of mathematics as well as a number of newer approaches that vary the logic and formal systems allowed in the first case and variations on computation complexity notions in the second. Finally, we present a glimpse of a new setting for the universe of reverse mathematics. It is an analysis the sort that has become common for recursion theoretic degree structures but yields drastically different results for proof theoretic strength.

**Lecture 2 of 2: **(April 26, 2023 @ 4:10–5:00 pm, Location: 60 Evans Hall):

We view the structure of reverse mathematics as a degree structure similar to that of the Turing degrees,